Problem: Simplify the following expression: $ q = \dfrac{a + 9}{-2a + 2} - \dfrac{-9}{7} $
In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{7}{7}$ $ \dfrac{a + 9}{-2a + 2} \times \dfrac{7}{7} = \dfrac{7a + 63}{-14a + 14} $ Multiply the second expression by $\dfrac{-2a + 2}{-2a + 2}$ $ \dfrac{-9}{7} \times \dfrac{-2a + 2}{-2a + 2} = \dfrac{18a - 18}{-14a + 14} $ Therefore $ q = \dfrac{7a + 63}{-14a + 14} - \dfrac{18a - 18}{-14a + 14} $ Now the expressions have the same denominator we can simply subtract the numerators: $q = \dfrac{7a + 63 - (18a - 18) }{-14a + 14} $ Distribute the negative sign: $q = \dfrac{7a + 63 - 18a + 18}{-14a + 14}$ $q = \dfrac{-11a + 81}{-14a + 14}$ Simplify the expression by dividing the numerator and denominator by -1: $q = \dfrac{11a - 81}{14a - 14}$